Elements of quantum statistics

Because of mutual influence, some atomic electrons are generalized forming a gas of quasi-electrons in crystals. These electrons preserve some properties of free electrons (e.g., each of them possesses a classical momentum) but, at the same time, also possess properties that distinguish them from really free particles (e.g., they have a mass different from the classical electron mass). Some well-known electric properties of metals are caused by this gas. However, it appears that in a model of free electron gas theoretical calculations strongly overestimate experimentally known characteristics only a small part of the generalized electrons can take part in the formation of these properties. 

Are these electrons indeed the gas of free electrons The answer to this question has already been obtained no, there are no free electrons in metals electrons form energy bands that can either be overlapping, or be separated by the forbidden energy gap. Besides, moving in a periodic crystal field, electrons are symmetry-dependent. It is necessary to answer one more question how are electrons distributed among these bands and how are they distributed inside the band  

In classical Newtonian physics the elementar volume of a configurational space cell is infinitesimal (it looks like Planck s constant ti is accepted to be zero) the electron distribution upon their energy is given by Maxwell-Boltzmann statistics there are large amounts of particles, all of which tend to occupy the state with the lowest energy, though chaotic temperature motion, on the other hand, scatters them on different energies. This process is described by the Boltzmann factor. 

It is suggested in quantum statistics that particles with half-integer spin obey Fermi-Dirac statistics such particles are called fermions. Electrons have a spin quantum number equal to 1/2 and therefore are fermions and must obey Fermi-Dirac statistics. 

The mathematical description of Fomi-Dirac statistics is given by the Fermi distribution function 

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Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

Calculation of the contributions of rotation and translation involves the use of quantum statistics, but to obtain a numerical solution the quantum statistics are usually replaced by classical statistics at temperatures above about 10 K below 10 K this classical approximation no longer holds. For this reason the equations presented here fail in the vicinity of 0 K. In agreement with the third-law concept, C° and are zero at 0 K. For a reference element, log A f is zero at 0 K, while for compounds the absolute values of the Gibbs energy function and log become infinite at 0 K, for the choice of the enthalpy reference temperature of 298.15 K. 

D. ter Haar, Elements of Statistical Mechanics, Holt, Rinehart, and Winston, New York, 1954 W. Band, Introduction to Quantum Statistics, D. Van Nostrand Co., Inc., Princeton, N.J., 1955. 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

Because 0 <. S, , < 1, the elements of S are completely analogous to the basis-set overlap integrals familiar in quantum chemistry (50). As the off-diagonal elements of the matrix are non-zero, that is, S , 0 for all i /, the basis is non-orthogonal. In some applications S is equivalent to what is typically called the metric matrix in statistics S is equivalent to the correlation matrix. 

If the activities of the laboratory in this field are said to be at the borders of quantum chemistry and statistical thermodynamics, these two disciplines are declared to be techniques." The problems raised by molecular liquids and solvent effects can be solved, or at least simplified by these techniques. This is firmly stated everywhere the method of calculation of molecular orbitals for the o-bonds was developed in the laboratory (Rinaldi, 1969), for instance, by giving some indications about the configuration of a molecule. The value and direction of a dipolar moment constitutes a properly quantum chemistry method to be applied to the advancing of the essential problems in the laboratory. In the same way, statistical mechanics or statistical thermodynamics constitute methods that were elaborated to render an account of the systems studied by chemists and physicists. In Elements de Mecanique Statistique, these methods are well said to constitute the second step, the first step being taken by quantum chemistry that studies the stuctures and properties of the constitutive particles. [53]... 

But the relative strengths ofatomic lines differ from star to star. The confluence of atomic physics, of quantum mechanics, and of statistical mechanics has allowed astronomers to understand these variations in detail. These issues were at the heart of the revolution that was 20th-century physics but today they are understood. The net resultis that other stars have different abundances of the elements than does our own. Perhaps one should say modestly different. The broad comparisons between the elements remain valid - iron is quite abundant, vanadium is rather rare. That remains true but many stars have many fewer of each. A few have more of each. This was a great discovery of 20th-century astronomy, because it established the nucleosynthesis of the elements as an observational science. Astronomers also learned how old the stars are, for there do exist telltale signs of a star s age. The oldest stars are found to have many fewer of all chemical elements (except the three lightest elements) than does the Sun. These came to be called metal-poor stars, because the heavy elements were lumped together under the term metals by astronomers. It may seem paradoxical that the oldest stars have the fewest metals but the key is that the abundances within... 

Two other, partially successful models to account for elemental periodicity, were proposed before and forgotten after the advent of quantum theory. An anonymous proposal, later ascribed to Prout, was based on the assumption that all atoms are composites of hydrogen. The purpose of this proposal was to account for the statistically improbable distribution of relative atomic weights, close to integer values. Following the discovery of isotopes Prout s hypothesis gained some new respectability, but it has never been fully exploited. Another theory was summarized by its author [20] in the statement ... 

Clementi (1985) described ab initio computational chemistry as a global approach to simulations of complex chemical systems, derived directly from theory without recourse to empirical parametrizations. The intent is to break the computation into steps quantum mechanical computations for the elements of the system, construction of two-body potentials for the interactions between them, statistical mechanical simulations using the above potentials, and, finally, the treatment of higher levels of chemical complexity (e.g., dissipative behavior). This program has been followed for analysis of the hydration of DNA. Early work by Clementi et al. (1977) established intermolecular potentials for the interaction of lysozyme with water, given as maps of the energy of interaction of solvent water with the lysozyme surface. 

The SMx aqueous solvation models, of which the most successful are called AM1-SM2,27 AMl-SMla, and PM3-SM3, °- adopt this quantum statistical approach, which takes account of the ENP and CDS terms on a consistent footing. The NDDO models employed are specified as the first element (AMI or PM3) of each identifier. It is worth emphasizing that the SMx models specifically calculate the absolute free energy of solvation—a quantity not easily obtained with other approaches. We have reviewed the development and performance of the models elsewhere.203 We anticipate our further observations later in this chapter by noting that the mean unsigned error in predicted free energies of solvation is about 0.6-0.9 kcal/mol for the SMx models for a data set of 150 neutral solutes that spans a wide variety of functionalities. A number of examples are provided later in this chapter. 

In order to make the theory useful it is necessary to know the constant of proportionality, which is calculated in such a way as to give the classical limit of the number of quantum states. This matter is dealt with in standard books on statistical mechanics [26]. The result is that for a system with n degrees of freedom, i.e. n position coordinates q and n momentum coordinates p, the number of states in the infinitesimal volume element rfq rfp is equal to rfq rfp//i", where n is Planck s constant. The association of a phase space volume /i with each quantum state can be thought of as a consequence of the uncertainty principle, which limits the precision with which a phase point can be specified in a quantum mechanical system. 

In order to resolve the outstanding issues of the quantum-classical transition, and study the control of entanglement and decoherence without the foregoing restrictions, we must venture into the domain of Quantum Complex Systems (QUACS), either consisting of a large number of inseparable elements or having many coupled degrees of freedom. Modern statistical physics copes with... 

The key then is to somehow calculate the probability with which a specific quantum state contributes to the average values. As far as thermal systems in thermodynamic equilibrium are concerned, this is the central problem addressed by statistical thermodynamics. W( therefore begin our discuasion of some core elements of statistical thermodynamics at the quantum level but will eventually turn to the classic limit, because the phenomena addressed by this book occur under conditions where a classic description turns out to be adequate. We shall see this at the end of this chapter in Section 2.5 where we introduce a quantitative criterion for the adequacy of such a classic description. 

Two additional conmiciits apply at tliis point. First, there is a conceptual difference between the probabilistic element in quantum and classic statistical physics. For instance, in quantum mechanics, the outcome of a measurement of properties even of a single particle can be known in principle only w ith a certain probability. In classic mechanics, on the other hand, a probabilistic element is usually introduced for many-particle systems where we would in principle bo able to specify the state of the system with absolute certainty however, in practice, this is not possible because we are dealing wdth too many degrees of freedom. Recourse to a probabilistic description within the framework of classic mechanics must therefore be regarded a matter of mere convenience. The reader should appreciate this less fundamental meaning of probabilistic concepts in classic as opposed to quantiun mechanics. 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

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